Preference Aggregation & Its Mechanism

7 months ago

Atiqur Rahman

11 minutes

Aggregation of Preferences

Aggregate in economics means a summary measure which describes about the market or the economy. Thus, aggregation is a collection or the gathering of things together. It is derived from the Latin word ad, meaning to and gregare, meaning herd. Thus, it literally means to herd or to flock.

When the total demand of individuals is being studied collectively, then it is known as aggregate demand. Thus, preference aggregation in social choice theory in economics is defined as forming a collective preference from the various sets of alternatives. When individual preferences are aggregated into a collective form to choose a preferred alternative, then it is known as aggregation of preferences.

In preference aggregation, various individuals express their preferences over a set of alternatives, and these preferences are then aggregated into a collective preference. When individual preferences are presented in the form of aggregation, then it is also called a social welfare function.

The result of classical theory and studies by various classical economists shows that in social choice theory, it is very difficult to aggregate the preferences of individuals under various kinds of conditions available in the economy.

During the past few years, preference aggregation has experienced a dramatic increase in attention from various scientists and economists. In social choice theory, preference aggregation is considered to be the heart of deciding about individual preferences.

In preference aggregation, several individual preferences are given a ranking of two or more social alternatives, and a single collective preference ranking is worked out from the various alternatives. The rule for aggregating individual preferences is also known as the group preference aggregation rule. This rule is based on the values of individuals, which are measurable and can be used under conditions of certainty.

How individual preferences are aggregated is a major branch of study for various economists, but the most popular study for aggregating preferences is Arrow’s impossibility theorem. This is also known as Arrow’s paradox in economics.

Kenneth Arrow found out the rule for constructing social preferences from individual preferences. This theorem says that when voters have three or more than three options, then there is no ranking order voting system which can convert the preferences of individuals who are being ranked into a community-wide ranking system if it also meets the pre-specified set of standards. These pre-specified standards are also known as non-dictatorship.

This theorem is mostly applied in voting procedures for individuals. This theorem also says that until and unless there are mandatory principles for voting, it would be very difficult to have a clear order of individual preferences. This paradox is based on an ideal voting structure that should meet certain specific criteria like Pareto efficiency.

This theorem can be explained with the help of the following example. There are three candidates, A, B, & C, and these three candidates want to rank their preferences.

55 votes A>B>C (55 people prefer A over B and prefer B over C).
50 votes B>C>A (50 people prefer B over C and prefer C over A).
40 votes C>A>B (40 people prefer C over A and prefer A over B).

The conclusion is that candidate A has more votes, so he is the winner. B was not in any majority. C could be the winner as more people prefer C over A. Thus, this is the result of the Arrows theorem.

The other scholars who were associated with aggregating preferences were Condorcet and Bellman. Condorcet anticipated a key theme in modern social choice theory. Condorcet concluded that the majority system is subject to some surprising problems in social choice theory.

Fuzzy logic by Bellman also helps in solving the problem of preference aggregation. This model is suitable when human preferences cannot be estimated in numeric value because the binary values are not suitable for real-world conditions.

Thus, from time to time, various theories have been developed, and an extensive amount of research has been done by various economists to aggregate individual preferences.

Mechanism of Preferences Aggregation

Mechanisms are those decision-making functions which map individual preferences and put them in an order, and try that a possible outcome should be determined from the ordering procedure. It is also the art of designing various rules and regulations so that a desirable outcome is achieved, though each individual acts for his own self-interest.

For example, voting, protocols and auctions come under this category. The most common mechanism for aggregating individual preferences is majority voting.

(1) Majority Voting:

This is one of the most popular and common mechanisms in social decision theory. In this system, individual preferences are being voted on, and the option that receives the maximum majority of votes is chosen. The majority voting system should satisfy three criteria:

(a) Dominance:

If one option is being chosen by all the voters, then the aggregation must yield this choice.

(b) Independence of irrelevant alternatives:

If an independent alternative is also being introduced then also it will not change the preference between two choices.

(c) Transitivity:

This means that choices must satisfy the criteria of transitivity. For example, A>B and B>C, then A>C.

The concept of majority voting can be explained with the help of the following example:

There is a town which is deciding about for government school funding and taxes on education. Now, there can be three possibilities: high spending, medium spending and low spending. There are 3 groups which are represented in equal order: low-income parents, medium-income parents and high-income parents.

The result would be as follows:

Low-income parents have low incomes so they will send their children to public schools.
High-income parents are the opposite in nature, as their income is good, so they will send their children to private schools and will not take an interest in sending them to government schools.
Middle-income parents belong to middle-class society. Their income is medium, so they want moderate taxes. They cannot pay much from their pockets. So they will prefer that they send their children to a private school which is not as costly as compared to other private schools in the town.
The majority voting system does not work in the case of M Condorcet Paradox because the assumption in this paradox is that majority voting does not lead to a stable outcome.
Taxes are assumed to be non-regressive in nature.

Majority voting is also explained with the help of the following table:

Preference Ordering	Low	Middle	High
First	H	M	L
Second	M	L	M
Third	L	H	H

This table implies that L>H and M>L, which means that M>L>H. This table also reflects that H-L=1:2, H-M=1:2, and M-L=2:1. These results are reflected when majority voting delivers a consistent outcome.

In the case of the Condorcet Paradox, the majority voting will not deliver a consistent outcome then the table will be presented in the following way.

Preference Ordering	Low	Middle	High
First	H	M	L
Second	M	L	H
Third	L	H	M

This table implies that H-L=1:2, H-M=2:1, and M-L= 2:1, which states that L>H and M>L which can be expressed in the following way: M>L>H.

Condorcet Ballot was given in 1785, and this ballot focuses on face to face election of the candidate. Condorcet ballot is used to show the comparison of the candidates and which candidate is considered to be better than all the other candidates present in a face-to-face election.

Condorcet Ballot is also explained with the help of the following table.

Option	A	B	C	D	E	F
No of voters	5	4	2	6	8	2
1	X	X	T	T	Z	T
2	Y	Z	Y	Y	Y	Z
3	Z	Y	X	Z	X	Y
4	T	T	Z	X	T	X

This table reflects that x v/s y has 9 votes, y v/s x has 18 votes, y wins over x. Similarly, z beats x, y and t, x beats t, and y beats t, and therefore z is the winner.

There is one more voting procedure, which was given by Borda in 1783. Borda Ballot is based on the principle of the points system, which is explained with the help of the following table.

Option	A	B	C	D	E	F
No of voters	5	4	2	6	8	2
1	X	X	T	T	Z	T
2	Y	Z	Y	Y	Y	Z
3	Z	Y	X	Z	X	Y
4	T	T	Z	X	T	X

This table reflects that one gives 4 points to a candidate in the first position. 3 points to a candidate in a second position, 2 points to a candidate in a third position. The candidate with the highest score is elected. This table also shows that x has 64 points, y has 75 points, z has 74 points, t has 57 points, and y is the winner from the voting.

The conclusion from the Condorcet ballot and Borda ballot is that there are 2 different voting procedures and 2 different winners from these voting procedures.

(2) Median Voter Theory:

Preferences, if are single-peaked, then only the majority system will give a stable outcome. The median voter theory concept is applied when voters’ tastes fall under the middle set of categories.

The disadvantage of this theory is that sometimes it proves to be inefficient and does not give a clear picture of the intensity of preferences. This theory can also be explained with the help of the following example.

There is a town which is considering building a children’s park in the town. There are 1000 voters in the town. The park will cost 40,000Rs, which will be financed by a tax of Rs40 on each voter. 5oo people are willing to contribute to the building of the children’s park by donating RS 100 each.

500 of the voters are not interested in contributing a single amount.

Thus marginal benefit of the children park 500(100) + (5000) =50,000Rs

Marginal cost of building the children’s park is Rs 40,000.

The result is that it is optimal to build the children’s park as the marginal benefit is more than the marginal cost. In this case, 500 people are saying yes, and 500 people are saying no, so majority voting will not work under this system as consumers have a medium taste. According to social efficiency, the park should be built for the children.

Preference Aggregation & Its Mechanism

Aggregation of Preferences

Mechanism of Preferences Aggregation

(1) Majority Voting:

(a) Dominance:

(b) Independence of irrelevant alternatives:

(c) Transitivity:

(2) Median Voter Theory:

Read More in: Theory of Public Finance

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Aggregation of Preferences

Mechanism of Preferences Aggregation

(1) Majority Voting:

(a) Dominance:

(b) Independence of irrelevant alternatives:

(c) Transitivity:

(2) Median Voter Theory:

Read More in: Theory of Public Finance

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